document.write( "Question 1171151: How many solution does the equation $\"+x_1%2Bx_2%2Bx_3%2Bx_4=1+\"$ in integers between $\"+-3+\"$ and $\"+3+\"$ ? \n" ); document.write( "

Algebra.Com's Answer #796299 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

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\n" ); document.write( "The student responded to my original response saying that he wrote a computer program to find the correct answer, which he says is 224.

\n" ); document.write( "My original response was an attempt to show a solution using logical analysis and mathematics -- not a computer.

\n" ); document.write( "My approach was valid; but my counting of the number of ways was flawed. I have corrected those numbers and have come up with a new answer that agrees with his.

\n" ); document.write( "I have corrected my response below....

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\n" ); document.write( "I was waiting to see if another tutor would come up with a clever way to find the answer....

\n" ); document.write( "Not seeing that happening, here is the solution I came up with....

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\n" ); document.write( "Use a \"greedy\" algorithm, where you make an ordered list with the entries higher in the list using more of the larger numbers.

\n" ); document.write( "To make sure you don't duplicate entries in the list, require that the four numbers be listed in non-increasing order.

\n" ); document.write( "Then, for each of the sets of 4 numbers you find, determine the number of different arrangements of those 4 numbers.

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document.write( " pattern  requirement  4-tuples  # of arrangements\r\n" );
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document.write( "(3,3,a,b): a+b=-5: (3,3,-2,-3)     12\r\n" );
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document.write( "(3,2,a,b): a+b=-4: (3,2,-1,-3)    24\r\n" );
document.write( "                   (3,2,-2,-2)     12\r\n" );
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document.write( "(3,1,a,b): a+b=-3: (3,1,0,-3)     24\r\n" );
document.write( "                   (3,1,-1,-2)    24\r\n" );
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document.write( "(3,0,a,b): a+b=-2: (3,0,0,-2)      12\r\n" );
document.write( "                   (3,0,-1,-1)    12\r\n" );
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document.write( "(2,2,a,b): a+b=-3: (2,2,0,-3)     12\r\n" );
document.write( "                   (2,2,-1,-2)     12\r\n" );
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document.write( "(2,1,a,b): a+b=-2: (2,1,1,-3)      12\r\n" );
document.write( "                   (2,1,0,-2)     24\r\n" );
document.write( "                   (2,1,-1,-1)     12\r\n" );
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document.write( "(2,0,a,b): a+b=-1: (2,0,0,-1)      12\r\n" );
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document.write( "(1,1,a,b): a+b=-1: (1,1,1,-2)       4\r\n" );
document.write( "                   (1,1,0,-1)     12\r\n" );
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document.write( "(1,0,a,b): a+b=0:  (1,0,0,0)     4\r\n" );
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document.write( "        total number of ways:   224\r\n" );
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\n" ); document.write( "ANSWER: There are $\"cross%28118%29\"$ 224 different ordered 4-tuples of integers between -3 and 3 with a sum of 1.

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